Sub-cell approximations for viscoelastic flows—filament stretching

Abstract

The accuracy, stability and consistency of new stress interpolation schemes is investigated, based upon sub-cell approximations. This includes the contrast of two alternative hybrid spatial discretisations: a cell-vertex finite element/volume (fe/fv) scheme and a finite element equivalent (fe). Here, the interest is to explore the consequences of utilizing conventional methodology and to demonstrate resulting drawbacks in the presence of complex stress equation source terms. Alternative strategies worthy of consideration are presented for a constant shear viscosity model, that of Oldroyd-B, with strain-hardening and unbounded extensional properties. We demonstrate how high-order accuracy may be achieved by respecting consistency in our algorithmic constructions.Both fe-and fv-spatial discretisations are embedded within this methodology. Linear interpolation for stress, of either fe-or fv-form on triangular sub-cells, is referenced within parent triangular finite elements in two dimensions. Finite element discretization is employed for the momentum and continuity system, via a second-order pressurecorrection scheme. In this regard, a complex filament-stretching flow with a free-surface is selected to compute the transient evolution of kinematic and stress fields. Shortcomings of various up-winding schemes are discussed, whilst dealing with such free-surface type problems and appropriate strategies for dealing with these types of problems are outlined.

Modelling filament stretching flows with strain-hardening models and sub-cell approximations

“…This lies in stark contrast to our earlier approach, and that of others, which relied upon a surface height function description, h(x,t) [9]. There, the free-surface movement was purely radial.…”

“…Following reference [3] and to furnish equitable comparison, we have chosen in this work to limit attention throughout to a single filament aspect-ratio, 54 . 0 0 = Λ (see [9] for alternative choices).…”

“…[25,26]. This particular numerical algorithm is further modified to enhance temporal quality via an area-weighting procedure ( − T l α lhs version, see [9] for practical implementation detail).…”

“…The update to a given node l is obtained by summing contributions from its control volume l Ω , which is composed of all fv-triangles surrounding node l, see [9]. Flux and source residuals may be evaluated over different control volumes.…”

“…In the present study, we employ a novel hybrid finite volume/element scheme (h y -fV) developed in [9] and applied here specifically within the transient viscoelastic free-surface context. The scheme is centred about a number of key features.…”